support_eq_limsInf — Mathlib

English

The support of a summation filter equals the liminf of its image under Finset.toSet.

Русский

Поддержка равна liminf образа фильтра под Finset.toSet.

LaTeX

$$$ \text{support }(L) = \liminf (L.filter.map \mathrm{Finset.toSet})$$$

Lean4

theorem support_eq_limsInf (L : SummationFilter β) : support L = limsInf (L.filter.map Finset.toSet) :=
  by
  refine eq_of_forall_ge_iff fun c ↦ ?_
  simpa [support, limsInf, setOf_subset] using
    ⟨fun hL b hb x hx ↦ hL x <| hb.mp <| .of_forall fun c hc ↦ hc hx, fun hL x hx ↦
      singleton_subset_iff.mp <| hL _ <| by simpa using hx⟩

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